Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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6
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1answer
82 views

How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can proof that the two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and ${d\over dx}\ln x$, some uses the former ...
5
votes
1answer
43 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
3
votes
1answer
62 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
3
votes
1answer
58 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...
3
votes
1answer
51 views

What does it mean for the determnant of a matrix to be independent of the vector space

Explain what it means for the determinant of the matrix, representing an operator $F$, to be independent of the basis of the vector space. Prove this property of the determinant. I'm not exactly ...
3
votes
1answer
101 views

What is Absolute convergence?

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), (u*v)(3), ...
2
votes
1answer
28 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
2
votes
1answer
55 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
2
votes
1answer
149 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
2
votes
1answer
46 views

Axiom or Postulate?

In wikipedia we see that the words “axiom” and “postulate” are synonyms: “An axiom, or postulate, is a premise or starting point of reasoning”. But in A Friendly Introduction to Numerical ...
2
votes
1answer
48 views

Ring of rational functions for reducible variety

Let $X$ be some affine algebraic variety over $\mathbb{k}$ (i.e. some closed subset in $\mathbb{A}_\mathbb{k}^n$). First suppose $X$ to be irreducible. Then the algebra $\mathbb{k}[X]$ is a domain and ...
1
vote
1answer
116 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
1
vote
1answer
44 views

Can all real numbers be presented via a natural number and a sequence in the following way?

Is there (for each fixed base system with digits $0,1,\dots,m$) and then for each real number $r\in\mathbb R$, an integer $n\in\mathbb Z$ and a sequence $(a_i)_{i\in\mathbb{N}}$ with ...
1
vote
1answer
42 views

Why is a graph an ordered pair?

From the source of all knowledge a graph is an ordered pair G = (V, E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V Why ...
1
vote
1answer
57 views

Mathematics and Origami

I am reading through this paper about the math behind origami: http://www.math.washington.edu/~morrow/336_09/papers/Sheri.pdf However, I am getting confused with definitions 3.3 and 3.4. I am not sure ...
1
vote
1answer
82 views

Acceptable Definition for $\sqrt{a}$?

Is this an acceptable definition for $\sqrt{a}$, where $a \in \mathbb{R}$? If $a\geq 0, \sqrt{a} = b \in \mathbb{R}$ s.t. $b\geq 0, b^2 = a$. I'm proving some theorems involving $\sqrt{a}$, in the ...
1
vote
1answer
57 views

Is it legal to define a piecewise define a function like this?

I'm trying to piecewise define a function $h$ using two other functions $f$ and $g$. I want to use $h$ to draw conclusions on a certain set $T$ that's a union of two other sets $A$ & $B$. $ ...
1
vote
1answer
74 views

What is an operator norm?

I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this: $|||\Omega-\hat{\Omega} |||_2$ where ...
1
vote
1answer
149 views

$\Sigma_k^\text{P}$−SAT definition is not clear to me

I don't understand if by saying there are $k$ alternating quantifiers on the variables $x_1$,$x_2$...$x_k$, It means we quantify ALL variables (there are only $k$ variables in the SAT formula) or just ...
0
votes
1answer
15 views

Beginning Proof on functions and Sum of functions

I've been having a hard time with this proof because I do not know where to go from where I am. The proof we were assigned to in class is as follows: Let $f : \mathbb Z \rightarrow \mathbb Z$ be a ...
0
votes
1answer
39 views

Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
0
votes
1answer
35 views

What is the meaning of an algebra?

An algebra $A(*,\hat{} ,\sim)$ is said to be Boolean algebra if it satisfies some conditions...In this statement what is the meaning of starting word an algebra?
0
votes
1answer
56 views

Natural definitions of families of subgraphs

Cluster analysis is a vibrant area of applied mathematics, "used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics". A subfield ...
0
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0answers
44 views

What does the sentence “every element of $S$ has a unique colour” mean?

Does the statement mean that each element of $S$ has exactly one colour, or that no two elements of $S$ share the same colour? Or could either interpretation be valid, depending on the context?
0
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0answers
23 views

What does it mean that solution of DE has ''n zeros''?

It appears in the definition of disconjugate DE: A linear DE with constant coefficents on a $t$-interval $I$ is said to be disconjugate on $I$ if no solution ($\neq 0$) has $n$ zeros on $I$. Thank ...
0
votes
0answers
40 views

definition of derived algebra $[L,L]$ of a Lie algebra $L$

Definition of derived algebra of a Lie algebra $L$ is given by linear span of commutators $[x,y]$ for $x,y \in L$. but here why do we take linear span and why cant we just consider collection of all ...
0
votes
0answers
84 views

Simplicial Homology Group, Delta set/complex

I'am confused. I'm trying to learn something about homology groups, but unfortunately I'm lost in some of the definitions/notations. Is there any difference between a delta set and a delta complex? ...
0
votes
0answers
28 views

Definition of restriction of relation

let be $\mathcal{R}$ a binary relation on $X$, and $\mathcal{T}$ a binary relation on $Y$, $\mathcal{T}$ is restriction of $\mathcal{R}$ if: 1) $Y \subseteq X$ 2) $\forall a,b \in Y( a T b ...
0
votes
0answers
49 views

Formal and general definition of natural domain (natural set) of a function

Can anyone give me a (as much as possible) formal and general definition of natural domain of a function? Let's say that a function is a triplet $(X,Y,f)$ where $f \subseteq X \times Y$ such that ...
0
votes
0answers
103 views

Rudin's definition of continuity in terms of pre-images (inverse images). Is this simple function continuous or not?

I am reading W. Rudins book ``Principles of Mathematical Analysis''. I find it hard to exactly understand the definition of continuity in terms of pre-images. Rudins definition of a continuous ...
0
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0answers
53 views

Definition of a Turing Machine

Could someone explain the following definition of a Turing Machine? A Turing Machine $M$ is defined formally by a tuple $(\Sigma, Q, \delta)$ Where $\Sigma$ is a finite set representing the number ...
0
votes
0answers
47 views

About definition of inductive set (with sets or ur-elements)!!

---let $A$ a set, $A$ is inductive if $\emptyset \in A$ and $\forall x \in A (x^+ \in A) $ ---let $A$ a set, $A$ is inductive if $\emptyset \in A$ and $\forall Y \in A (Y^+ \in A) $ an example of ...
0
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0answers
46 views

Help with definitions

I need the definitions of 2 following things: What is a characteristic function of the unit disc in $\mathbb{R}^{2}$? What is a distribution is obtained by integrating a function over the unit ...
0
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0answers
71 views

$\mathbb{R}$ as set of Dedekind cuts on $\mathbb{Q}$

-- "let $A,B \in \mathbb{Q}$, $A < B$ if $A\leq B$ and $A \neq B$ -- "let $\preceq$ be a ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq $ if $\forall a ...
0
votes
0answers
40 views

Similarities and differences between Memoryless property and Markovian property

I want to ask about similarities and differences between Memoryless property (MLP) and Markovian property (MKP), which I've been curious since long time ago. According to what I have searched and ...
0
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0answers
26 views

Sense of Petrowsky

What does it mean that a problem is well-posed in the sense of Petrowsky? The paper I am reading "Exact Controllability, Stabilization, and Perturbations for Distributed Systems" by J.L. Lions refers ...
0
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0answers
19 views

definition of raw variable

i would like to clarify definition of raw variable,for instance,let us see following article http://www.psych.umn.edu/faculty/waller/classes/FA2010/Readings/rodgers.pdf where it is written that The ...
0
votes
0answers
108 views

Understanding Gray-Level Hit-or-Miss-Transform Definition

From a paper I am using for my Bachelor thesis: The HMT transform by the pair $(A,B)$ associates to a binary image $X$ the set $X\otimes(A,B)$ of positions where the translate of $A$ fits inside ...
0
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0answers
39 views

Definition: “A contour respects causality”

When doing a contour integral, what does "the contour respects causality" mean?
0
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0answers
43 views

Are my definitions of cotangent space, differential and differential forms and coboundary operator correct?

Define the cotangent space $T_a^*\mathbb{R}^n$. Define the differential of a dunction $f$ at the point $a, df \in T_a^*\mathbb{R}^n$. Write down the explicit formula for the deffertial $df$ in ...
0
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0answers
90 views

Analyticity and differentiability

May this is be an easy question. I have some problem with difference between consepts Analyticity and differentiability a complex function.
0
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0answers
57 views

Improper or Undefined

Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral $\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined? If we take it as a legitimate function for improper Riemann ...
0
votes
0answers
192 views

What is derived number definition? ( in Vitaly covering)

In Vitali covering definition i see "derived number" word, but I dont know what that mean. Example for vitali covering: If $f$ is strictly increasing and $$E=\{x: \text{ there is a derived number } ...
0
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0answers
122 views

What is the correct definition for an imaginary number?

The following is taken from Wikipedia's definition. An imaginary number is a number whose square is less than or equal to zero. But I also heard that An imaginary number is a number whose ...
0
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0answers
73 views

Definition of matrix derivative

Let $A$ be an $m \times n$ matrix. Let $a_{ij}$ be an element of $A$. What does the notation $\frac{\partial A}{\partial a_{ij}}$ mean?
0
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0answers
73 views

Definition of “eventually dominates”

What is the definition of the term "eventually dominates"? I guess it's either $f$ eventually dominates $g$ if for large enough $n$, $f(n) > g(n)$ or the same with $\ge$. A quick Google search ...
0
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0answers
52 views

Help in Analysis

I'm studying the article "On W1;p Estimates for Elliptic Equations in Divergence Form" of L. A. CAFFARELLI and I. PERAL. There, you can find We use the classical Hardy-Littlewood maximal operator, ...
0
votes
0answers
67 views

Concerning the point stabilizing group and coset stabilizing group.

I would like to know more about the point stabilizer group and the coset stabilizer group, like the definitions, why they are used in group theory, who developed them and there importance.
0
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0answers
173 views

Equinumerosity without ordered pairs

After some discouraging comments to this question let me ask straight ahead: Can the concept of equinumerosity be defined basically without the concept of ordered pairs (in any of its ...
-1
votes
0answers
34 views

Fixed point - definitions (asymptotically stable, repelling)

Given the following fixed point $(\widehat{x},\widehat{y} )$. If this point is not asymptotically stable, can I then conclude it is a repellor? Is a repellor/repeller defined to be a not ...